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First, if I have a 1/1 creature with first strike and deathtouch, and my opponent blocks it with a 2/2 creature, will my creature die? I can see two alternative ways this could play out and I'm not sure which is correct.

My creature deals his first strike damage, immediately killing the opponent's creature before he is able to deal combat damage as a state-based effect.

My creature deals his first strike damage, which would normally be lethal due to deathtouch, but state-based effects are not checked, and thus the opposing creature deals his damage and both die.

I'm almost positive in this case that it's the first option above.

What gets trickier is, for example, if a creature has first strike and a triggered ability "Whenever this creature deals combat damage to a creature, return that creature to its owner's hand."

This is not a state-based effect, but rather uses the stack. Similar to above, there are two possibilities.

My creature deals his first strike damage, and his triggered ability is placed on the stack. The stack waits to resolve until after all combat damage is dealt, so the opposing creature still kills mine.

My creature deals his first strike damage, and his triggered ability is placed on the stack. Both players now gain priority between first strike and combat damage (which I don't think normally happens, thus my confusion). The stack finishes resolving, and the opponent's creature doesn't have a chance to deal damage.

Dying from damage is a state-based effect, so if the game didn't check for them between first strike and normal damage, there wouldn't be any point to first strike - a first strike creature wouldn't be able to kill a normal damage creature before it hit back!
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JefromiFeb 24 '13 at 15:55

To end 510.4, both players must pass priority until everything on the stack resolves. When the stack is empty, play proceeds to the next step, which will be a second Combat Damage Step (if the first was for first strike) or the End of Combat Step (normally).

506.1. The combat phase has five steps, which proceed in order: beginning of combat, declare attackers, declare blockers, combat
damage, and end of combat. The declare blockers and combat damage
steps are skipped if no creatures are declared as attackers or put
onto the battlefield attacking (see rule 508.4). There are two combat
damage steps if any attacking or blocking creature has first strike
(see rule 702.7) or double strike (see rule 702.4).

If a creature has first strike, there are two combat damage steps instead of one.

As long as you just treat these two combat damage steps the same as you would any other step during the combat phase, I don't think you can go far wrong. You resolve everything that's on the stack before finishing the declare attackers step and moving on to declare blockers, right? From there it shouldn't be hard to see that you should also clear the stack before moving from the first strike combat damage step to the normal combat damage step. Therefore option number (2), not number (1), holds true in your second example.